Logarithmic negativity (LN)

[KINGSHUK ADHIKARY]

Hello,

I have a density matrix (see attached figure) for a bipartite system (qubit+oscillator). I want to calculate the LN of each subsystem separately. The qutip code "logarithmic_negativity(V, g=1.4142135623730951)" does not work for me because an error appears in covariance matrix V, like LinAlgError: Last 2 dimensions of the array must be square. Maybe I am taking the wrong steps. I am not sure why this error was generated here.

So, I want to calculate the LN from the definition of partial transpose, but I do not understand the command "partial_transpose(rho, mask, method='dense')" specifically the significance of mask.

I would appreciate it if anyone could suggest some valuable advice in this context.

Regards,

Kingshuk

[Alex Pitchford]

Hello Kingshuk

Yes, it seems like from your attachment that you have a covariance matrix. I am not so sure why you would want to partial_transpose. Maybe you want partial_trace instead? This should return the subsystem that you are interested in. Though given the dimensions, maybe you are working a Fock basis for the oscillator and may need to convert to covariance matrix first. I would expect a single oscillator to have a 2x2 covariance matrix.

I hope that helps (at least a bit). As a disclaimer, I know a little about Gaussian states, but not so much about the qutip continuous variable functions, but happy to try and learn a little more.

Alex

[Alex Pitchford]

Sorry, the 2x2 oscillator was assuming a Gaussian state. I think the partial_trace suggestion is still valid though. It is not that clear though why you would get LN for a separate system, as it is a measure of entanglement.

[KINGSHUK ADHIKARY]

Hello Alex,

Thanks for your attention. I have already calculated the partial trace to separate any state of my interest from a joint system and then calculated the Von-Neumann entropy. I need only a partial transpose to calculate the LN (entanglement measure) of my system, where the partial trace has no role. Though I have attempted to use the covariance matrix, I am getting the following error message: So far, I know that one can calculate LN in two distinct ways: (1) the partial transpose and (2) the covariance matrix. Unfortunately, I cannot reach LN from these formalisms.

 

I would be glad if you provided any leads about this subject.

Regards,

Kingshuk

[umar shifaqat]

I am also trying to calculate Logarithmic Negativity for my system. I am doing it in several different ways to check if any of the methods can work out. Currently, I am getting correct values (in some range). I am trying different ways to compute accurate values. One of the methods is adding absolute values of each colomn and finding max from these, then taking log.